1 Teletraffic Theory and Engineering Robert B. Cooper and Daniel P. Heyman I. INTRODUCTION A telecommunications network consists of expensive hardware (trunks, switches, etc.) with the function of carrying telecommunications traffic (phone calls, data packets, etc.). The physical network is fixed, but the traffic that it is designed to carry is random. That is, the times at which calls are generated are unpredictable (except in a statistical sense), and, similarly, the lengths of time that the calls will last are unpredictable;yet, the network designers must decide how many resources to provide to accommodate this random demand. If the resources are provided too sparingly, then the quality of service will be low (e.g., too many calls will be lost because the required resources are not available when needed);but, if the resources are provided too generously, then the costs will be too high. Teletraffic theory deals with the mathematical analysis of models of telecommunications systems and with the interrelationships among the provision of resources, the random demand, and the quality of service; teletraffic engineering addresses the art and science of the application of this theory to the design of real telecommunications systems. Encyclopedia of Telecommunications, Volume 16, Pages 453–483 Copyright n 1998 by Marcel Dekker, Inc., 270 Madison Ave., New York, New York All rights reserved. Publisher and Contributor hereby agree that the Entry has been specially commissioned by the Publisher for use as a contribution to a collective work and shall be and is considered a ‘‘work-made-for-hire.’’ Publisher shall own all right, title and interest in and to the Entry, including without limitation the entire copyright therein, and shall be deemed to be the author of the Entry for copyright purposes. Publisher shall be free to use the Entry, in whole or in part, in all languages, throughout the world, in perpetuity, in any form or medium now known or hereafter developed, and to license others to do the foregoing. 1 2 Cooper and Heyman II. HISTORY The telephone was patented in 1876, and the first commercial telephone switchboard went into operation in 1878 in New Haven, Connecticut. It consisted of a set of subscribers who could be connected two at a time via a single path. It has been said that the need for teletraffic theory became apparent as soon as the number of subscribers grew to three! The first significant advance in teletraffic theory came in 1917, when A. K. Erlang, a scientist/mathematician/engineer working for the Co- penhagen Telephone Company, published a paper that described a method and used it to derive some formulas that provide the basis for much of modern teletraffic theory and engineering. Later, with the invention of operations research during World War II, Erlang’s methods and models were incorporated into queueing theory, and these two subjects (queueing, teletraffic theory) are now closely intertwined. (A queue is a waiting line. Queueing theory is the mathematical theory of systems that provide service to cus- tomers with arrival times and service requirements that are random. If servers are unavailable to accommodate arriving customers, then a queue may form, hence the name.) There is now a huge amount of literature on queueing (and teletraffic) theory, and papers are being published in the technical journals at an ever-increasing rate. In this article, we survey basic teletraffic (and queueing) theory, and we discuss both classical applications and new theory for applications that are driven by recent advances in telecommunications technology and computer science. III. BASIC CONCEPTS We take as our basic model a system in which calls arrive at random times, and each call requests the use of a trunk. (In this article, we will ignore the distinction between trunks, which interconnect the switches, and lines, which connect the subscribers to the switches.) If a trunk is available, the call holds it for a random holding time, and if no trunk is available, the blocked call takes some specified action, such as overflowing, retrying, or waiting in a queue. (In queueing theory parlance, the calls are customers, the trunks are servers, and the holding time is the service time.) The objective of teletraffic theory is to derive appropriate descriptions of the random teletraffic (a description of the statistical, or stochastic, properties of the arrival times and holding times) and to derive formulas that describe the performance of the system (e.g., the probability of blocking, the fraction of calls that overflow, the average waiting time, etc.) as a function of the demand and the number of trunks. This theory is then adapted and applied to the design and administration of real telecommunications systems;that is teletraffic engineering. The central concept of teletraffic engineering is the stochastic nature of teletraffic, so the underlying mathematics used are probability, statistics, and stochastic processes. Therefore, we summarize (as briefly as possible) these mathematical processes. Then, we apply this to derive and understand the basic formulas of teletraffic theory. To make this theory concrete and to explore the robustness of these formulas, we describe briefly the essential concepts of simulation, and we give pseudocode that can be used to write computer programs to simulate these models. Teletraffic Theory and Engineering 3 A. Birth-and-Death Process To fix these concepts, consider the following model. Calls arrive according to a sto- chastic process (described below) at a group of s identical trunks. If an arriving call finds a trunk available, the call holds the trunk for a random holding time (described below), after which the call drops the trunk, which then becomes available for another call. If the arriving call finds all s trunks busy, then the call is blocked, in which case it takes some specified action (described below). Suppose that Pj denotes the long-run probability that the system is in state j, that is, Pj is the probability that the number of calls present (in service or, if the model permits, waiting in the queue for a trunk to become available) is j;assume that when the system is in state j, then the call arrival rate is Ej, and the call departure rate is lj. Then, it can be shown that, under certain conditions that must be satisfied by the arrival process and the departure process (discussed below), the following equations determine the state probabilities as a function of the rates Ej and lj: EjPj ¼ lj þ 1Pj þ 1 ðj ¼ 0; 1; 2; ...Þð1Þ and ::: P0 þ P1 þ ¼ 1 ð2Þ Equation (1), originally derived by Erlang, can be given the following inter- pretation: rate up from state j equals rate down from state j + 1. That is, the term EjPj on the left-hand side of Eq. (1) equals the fraction of time Pj that there are j calls present multiplied by the rate at which calls arrive when there are j calls present; hence, the product EjPj equals the long-run rate (in transitions per unit time) at which the system state jumps from level j to level j + 1. Similarly, the right-hand term lj+1Pj+1 equals the long-run rate (in transitions per unit time) at which the system state jumps down from level j + 1 to level j. Therefore, if the system is to be in ‘‘equilibrium,’’ Eq. (1) must hold. Successive solution of Eq. (1) for each Pj in terms of the previous ones gives ::: E0E1 EjÀ1 Pj ¼ ::: P0 ð3Þ l1l2 lj and P0 is calculated from the normalization condition, Eq. (2) (which simply requires the sum of the fractions of time that the system spends in each state to add to 100%). 1 P0 ¼ E0 E0E1 4 1 þ þ::: ð Þ l1 l1l2 A stochastic process that is described by Eq. (1) is called a birth-and-death process. The key technical point here is that the instantaneous rates Ej and lj are assumed to depend on only the present state and are otherwise independent of the past history of the process. The birth-and-death probabilities Pj ( j =0,1,...) defined by Eq. (1) are time-average probabilities;that is, Pj can be interpreted as the fraction of time that the system spends in state j. Also of interest are the customer-average probabilities Cj ( j =0,1,...); Cj can be interpreted as the fraction of customers arriving when the system is in state j. In general, the fraction of time that the system 4 Cooper and Heyman spends in a given state does not equal the fraction of customers finding that state when they arrive. However, when the customers arrive according to a Poisson process (defined below), then Cj ¼ Pj ðj ¼ 0; 1; ...Þð5Þ The important equality Eq. (5) reflects the PASTA theorem (Poisson Arrivals See Time Averages). Sometimes, the P’s are called the outside observer’s distribution (reflecting the notion that they measure the frequencies of occurrence of the states as seen by an outside observer passively observing the system continuously or at random instants), and the C’s are called the arriving customer’s distribution (reflecting the notion that they measure the frequencies of occurrence of the states as seen by the arriving customers). The PASTA theorem says that, remarkably, a stream of Poisson arrivals will see the states with the same frequencies as will an outside observer, even though the arrivals, in general, ‘‘cause’’ the states of the system and view the system just prior to the instants of upward state transitions, whereas the outside observer has no causal effect on the states of the system. (There are some situations for which non- Poisson arrivals see time averages, but these are rather special.) We now discuss how these results are applied to our basic teletraffic model.